Calculate the given quantity if u = i + j − 5k v = 4i − 5j + k w = j − 4k.

Question

Calculate the given quantity if

u = i + j − 5k    v = 4i − 5j + k    w = j − 4k.

(a)    2u + 3v

(b)    |u|

 

(c)    u · v

 

(d)    u × v

 

(e)    |v × w|

 

(f)    u · (v × w)

 

(g)    the angle between u and v (rounded to the nearest degree)

 °

Answer 1

Hi,

u = i + j − 5k    v = 4i − 5j + k    w = j − 4k.

(a)    2u + 3v = 2(i + j − 5k  ) + 3 (4i − 5j + k) = 14i -13j-7k

(b)    |u|=√1 +1 + (-5)^2 = √27

(c)    u · v= ( i + j − 5k).(4i − 5j + k ) = 4 -5-5 = -6

(d)    u × v = |i j k|

|1 1 -5| = i ( 1-2 5 ) -j (1+20) +k(-5-4) = -24i -21j-9k 0

|4 -5 1|

(e)    |v × w| first find    v × w ( the same method of u x v) Then use (b)

(f)    u · (v × w) now you have    v × w so use dot product between   u · (v × w) looks like (c)

(g) the angle between u and v (rounded to the nearest degree) ----> cos θ = u.v/[||u||.||v||]

you know ||u|| = √27 ||v||= √16+25+1 =√42 Now cos θ = -6/√27.√42 = -0.17817 =.> θ = 100.26

I hope it is useful,

Minoo